Students (and teachers) are often fascinated by the fact that certain
geometric images have fractional dimension. The Sierpinski triangle
provides an easy way to explain why this must be so.

To explain the concept of fractal dimension, it is necessary to
understand what we mean by dimension in the first place. Obviously, a
line has dimension 1, a plane dimension 2, and a cube dimension 3. But
why is this? It is interesting to see students struggle to enunciate
why these facts are true. And then: What is the dimension of the
Sierpinski triangle?

They often say that a line has dimension 1 because there is only 1 way
to move on a line. Similarly, the plane has dimension 2 because there
are 2 directions in which to move. Of course, there really are 2
directions in a line -- backward and forward -- and infinitely many in
the plane. What the students really are trying to say is there are 2
linearly independent directions in the plane. Of course, they are
right. But the notion of linear independence is quite sophisticated
and difficult to articulate. Students often say that the plane is
two-dimensional because it has ``two dimensions,'' meaning length and
width. Similarly, a cube is three-dimensional because it has ``three
dimensions,'' length, width, and height. Again, this is a valid
notion, though not expressed in particularly rigorous mathematical
language.

Another pitfall occurs when trying to determine the dimension of a
curve in the plane or in three-dimensional space. An interesting
debate occurs when a teacher suggests that these curves are actually
one-dimensional. But they have 2 or 3 dimensions, the students object.

So why is a line one-dimensional and the plane two-dimensional? Note
that both of these objects are self-similar. We may break a line
segment into 4 self-similar intervals, each with the same length, and
ecah of which can be magnified by a factor of 4 to yield the original
segment. We can also break a line segment into 7 self-similar pieces,
each with magnification factor 7, or 20 self-similar pieces with
magnification factor 20. In general, we can break a line segment into
N self-similar pieces, each with magnification factor N.

A square is different. We can decompose a square into 4 self-similar
sub-squares, and the magnification factor here is 2. Alternatively, we
can break the square into 9 self-similar pieces with magnification
factor 3, or 25 self-similar pieces with magnification factor 5.
Clearly, the square may be broken into N^2 self-similar copies of
itself, each of which must be magnified by a factor of N to yield
the original figure. See Figure 8. Finally, we can decompose a cube
into N^3 self-similar pieces, each of which has magnification factor
N.

Figure 8: A square may be broken into N^2
self-similar pieces, each
with magnification factor
N

Now we see an alternative way to specify the dimension of a
self-similar object: The dimension is simply the exponent of the
number of self-similar pieces with magnification factor N into which
the figure may be broken.

So what is the dimension of the Sierpinski triangle? How do we find
the exponent in this case? For this, we need logarithms. Note that,
for the square, we have N^2 self-similar pieces, each with
magnification factor N. So we can write

Similarly, the dimension of a cube is

Thus, we take as the definition of the fractal dimension of a
self-similar object

Now we can compute the dimension of S. For the Sierpinski triangle
consists of 3 self-similar pieces, each with magnification factor 2.
So the fractal dimension is

so the dimension of S is somewhere between 1 and 2, just as our
``eye'' is telling us.

But wait a moment, S also consists of 9 self-similar pieces with
magnification factor 4. No problem -- we have

as before. Similarly, S breaks into 3^N self-similar pieces with
magnification factors 2^N, so we again have

Fractal dimension is a measure of how "complicated" a self-similar
figure is. In a rough sense, it measures "how many points" lie in a
given set. A plane is "larger" than a line, while S sits somewhere
in between these two sets.

On the other hand, all three of these sets have the same number of
points in the sense that each set is uncountable. Somehow, though,
fractal dimension captures the notion of "how large a set is" quite
nicely, as we will see below.